2.5 Determining the reflectivity of an AR coating on a QCL facet
2.5.4 Model for photon flux density distribution in saturated amplifier with
To generalize the usual form of laser behavior discussed in Section 1.4.1, we start at a more basic level than is given by Eq. (1.2) and develop a model without making some of the simplifications that lead to Eq. (1.2). This model uses elements of the theory of fiber lasers and amplifiers, the expressions in use can be found in many text books, for instance [133]. However, to keep to a level of complexity well-fit for our purposes, any phase-dependence of the electric field is omitted in this discussion, and photon flux densities that are proportional to E2 are considered instead. However, the phase information can be added back in to arrive
at expressions similar to Eq. (1.6), but the resulting expressions are far more tedious to write down or to compute. A more detailed discussion of this will be given in Sections 2.5.7 and 2.5.8.
Consider a general laser with an upper and a lower laser level. The probability of absorption of a photon by an electron in the lower level per second is
W = ϕσ(ν), (2.40)
where ϕ is the photon flux density (in photons per unit area per unit time) and σ is the transition cross section
σ(ν) = λ
2
8πtspc
l(ν), (2.41)
where ν = 1/λ is the wavenumber of the light, λ is the wavelength, tsp is the spontaneous
emission lifetime of the transition and l(ν) is the normalized lineshape function of the tran- sition (of width ∆ν and peak value of 2
π∆ν). The probability for stimulated emission by an
electron in the upper laser level is the same as Eq. (2.40).
If N1 is the density of electrons in the lower state (in electrons per unit volume) and N2
is the density of electrons in the upper state, then the rate of photons generated through stimulated emission (per unit volume) is N2W and the rate of photons absorbed is N1W.
We call N = N2− N1 the population difference and if N > 0, there is population inversion,
will release more photons traveling in the z direction through stimulated emission than are absorbed and – if spontaneous emission is ignored for now – the net rate of gained photons in an infinitesimal volume is just the rate of photon emission minus the rate of photon absorption.
dϕ
dz = (N2− N1)W = NW = Nϕ(z)σ(ν) = ϕ(z)γ(ν), (2.42) where we have introduced the gain
γ(ν) = Nσ(ν), (2.43)
where the unit of the gain is 1/cm. The light intensity is related to the photon flux density through
ˆI(ν) = hcνϕ(ν), (2.44)
where h is Planck’s constant and c is the speed of light in vacuum.
It is clear that every absorption event reduces N1 and ϕ by one and increases N2 by one.
On the other hand, every stimulated emission event reduces N2 by one and increases N1 and
ϕ. So there will be a balance between N1, N2, ϕ, at every given pumping rate ˆR. Calculation
of this balance is straight-forward with the use of the rate equations dN2 dt = ˆR2− N2 τ2 − N2W + N1W (2.45) and dN1 dt = − ˆR1− N1 τ1 + N2 τ21 + N2 W − N1W, (2.46)
where ˆR1 and ˆR2 are the pumping rates into levels 1 and 2, τ1 and τ2 are the overall lifetimes
of carriers in levels 1 and 2 due to both radiative and non-radiative decay and 1/τ21 is the
rate of transitions from level 2 to 1. Figure 2.11 shows a sketch of the situation.
Figure 2.11 The population densities N1 and N2 (cm−3s−1) of carriers in energy levels 1 and 2 are
determined by three processes: decay (at rates 1/τ1 and 1/τ2, respectively, which includes the effects
of spontaneous emission), depumping and pumping (at rates ˆR1 and ˆR2, respectively), and absorption
and stimulated emission (at rate Wi with corresponding time constant 1/Wi). Figure taken from [133]
These equations can be solved for the steady-state (dN2
dt = 0 and dN1
dt = 0), using N =
N = N0 1 + τsW (2.47) where τs = τ2+ τ1(1 − τ2 τ21) (2.48) and N0 = ˆR2τ2(1 − τ1 τ21) + ˆ R1τ1. (2.49)
N0 is the steady-state solution to N in the absence of radiation (W = 0).
Substituting Eq. (2.47) and Eq. (2.40) into Eq. (2.43) gives
γ(ν) = γ0(ν)
1 + ϕ/ϕs(ν) (2.50)
where
γ0(ν) = N0σ(ν) (2.51)
is the gain in the absence of amplifier radiation and ϕs= τsσ(ν)1 is called the saturation photon
flux density. γ0(ν) is the gain at the very beginning of the pumping process, where population
inversion has been established through rapid pumping but not enough light has been produced by spontaneous emission yet to seed stimulated emission. γ0(ν) is therefore called the small-
signal gain. As is clear from Eq. (2.50), as the photon flux ϕ inside the resonator grows, the overall gain γ decreases, thus γ is called the saturated gain. When ϕ = ϕs, the overall gain
has decreased to half its value for the empty resonator, γ0.
From Eq. (2.50) it is clear that the gain is a function of photon flux, thus Eq. (2.42) becomes
dϕ
dz = ϕ(z)
γ0(ν)
1 + ϕ(z)/ϕs(ν) (2.52)
and thus the gain is also a function of z, thus, the solution to Eq. (2.50) is not a simple exponential function as it may seem from Eq. (2.43) due to gain saturation. Even in the regime of very little light it is not, since Eq. (2.50) still ignores the presence of spontaneous emission, which in the limiting case of very little stimulated light is not a good approximation. But this will be resolved soon.
Absorption is not the only way to lose photons, but there are other loss mechanisms. The first one to be treated is the scattering of photons out of the system. Introducing the phenomenological distributed waveguide loss αw (in units 1/cm), the propagation equation
of the photon flux density becomes dϕ
Equation (2.53) only considers the photons traveling in the positive z-direction, but natu- rally there are also photons traveling in the negative z-direction. Making the obvious gener- alization, Eq. (2.53) splits up into two equations
dϕ1
dz = ϕ1(z)(γ(ν, z) − αw) (2.54)
and
dϕ2
dz = −ϕ2(z)(γ(ν, z) − αw), (2.55)
where ϕ1(z) now denotes the photons traveling in the positive z direction and ϕ2(z) denotes
the photons traveling in the reverse direction. These equations are coupled since γ(ν, z) is saturated by photons traveling in both directions, thus Eq. (2.50) becomes
γ(ν, z) = γ0(ν) 1 +ϕ1(z)+ϕ2(z)
ϕs(ν)
(2.56)
and with this, Eq. (2.54) and (2.55) become
dϕ1 dz = ϕ1(z) γ0(ν) 1 +ϕ1(z)+ϕ2(z) ϕs(ν) − αw (2.57) and dϕ2 dz = −ϕ2(z) γ0(ν) 1 +(ϕ1(z)+ϕ2(z) ϕs(ν) − αw . (2.58)
The next step is to insert a phenomenological term for the spontaneous emission, ξsp, whose
role will be more closely examined later. With this, Eq. (2.57) and (2.58) become
dϕ1 dz = ϕ1(z) γ0(ν) 1 +ϕ1(z)+ϕ2(z) ϕs(ν) − αw + ξsp (2.59) and dϕ2 dz = −ϕ2(z) γ0(ν) 1 +(ϕ1(z)+ϕ2(z) ϕs(ν) − αw − ξsp. (2.60)
To solve these equations, a set of boundary conditions is required. These are connected to the last loss mechanism for photons. At the (partially reflective) facets at each end of the waveguide, the light impinging on them generally splits up into two parts, the light that is transmitted and lost from the resonator, this is the useful light, and a part that is reflected back into the resonator, seeding the amplifier in the reverse direction. Thus the boundary conditions are
and
ϕ1(0) = R1ϕ2(0), (2.62)
where l is the length of the wave guide (resonator), R1 is the reflectivity of the facet at z = 0,
and R2 is the reflectivity of the facet at z = l.
For a given set of parameters αw, ϕs(ν), γ0(ν), and ξsp, the boundary value problem
consisting of Eq. (2.59)-(2.62) can easily be solved numerically, for instance with use of the Shooting Method, yielding self-consistent solutions for the photon density distributions ϕ1(z)
and ϕ1(z) for z ∈ (0, l). These solutions can now be related to observable output powers P1
and P2 from facets 1 and 2, respectively, through
P1= ϕ2(0)(1 − R1)Aℏω (2.63)
and
P2= ϕ1(l)(1 − R2)Aℏω, (2.64)
where A is the facet area and ℏω is the energy of a photon. At this point, it is not yet clear, how the laser parameters αw, ϕs(ν), γ0(ν), and ξsp are to be determined for a realistic laser.
This will be discussed in the following.
A laser can be viewed simply as a device that produces light when supplied with energy through pumping. In most semiconductor lasers, pumping is done by applying a voltage and running an electric current through the structure. The laser parameters now determine the response of the device to this pump current I. Thus it is a viable assumption that (most of) the laser parameters can be derived from the P-I curve, i.e. the response of output power P to pump current I.
To introduce the pump current into the problem, the laser parameters have to be brought in connection with it. Now while αw, ϕs(ν) are quite obviously, with sufficient accuracy,
invariant to current changes, reasonable ansatzes have to be made for γ0(ν) and ξsp. These
are the following. In interband lasers, the small-signal gain coefficient γ0(ν) is proportional
to the pump rate ˆR. The pump rate, however, over a wide range is proportional to the pump current. Although this range is far smaller for QCLs than it is for interband lasers, it is reasonable to state that even in QCLs, over any range of interest (where light is actually emitted), this is a good approximation. Thus
γ0(ν) ≈ ¯gI (2.65)
where ¯g is a new parameter, the gain coefficient that is constant with respect to current. The spontaneous emission factor ξsp is proportional to the population density of electrons
in the upper laser state, N2. The first assumption to be made now is that the lower laser state
mainly gets filled by electrons from the upper laser state, thus ignoring thermal backfilling and accidental non-radiative relaxation into the lower state from any state other than the
upper laser state (e.g. from the continuum). The second assumption is that the fraction of electrons from the upper laser state that reaches the lower laser state is constant with respect to current. If these assumptions are satisfactorily met, we can say the population density of the lower laser state N1is proportional to N2. But then N2 ∝ N2−N1= N. But since N ∝ γ,
according to Eq. (2.43), N ∝ I if we disregard gain saturation, which is not a problematic assumption, given that spontaneous emission is most important in ranges where there is little light and thus the gain is hardly saturated. But this means
ξsp≈ ˆξspI, (2.66)
where ˆξsp now is a parameter that is constant with respect to current.
These parameters can now be inserted into Eqs. (2.59) and (2.60) to give the full problem in terms of the pump current I as
dϕ1 dz = ϕ1(z) ¯gI 1 +ϕ1(z)+ϕ2(z) ϕs(ν) − αw + ˆξspI (2.67) and dϕ2 dz = −ϕ2(z) ¯gI 1 +(ϕ1(z)+ϕ2(z) ϕs(ν) − αw − ˆξspI. (2.68)
These equations in conjunction with an experimental P-I curve (P1(I) and P2(I)) deter-
mine the boundary value problems of the photon flux densities for all currents I. Now for any one given current, the boundary value problem is highly under-determined in terms of the parameters ¯g, ϕs(ν), αw, ˆξsp, R1, and R2, consequently there is a manyfold in the space
spanned by these parameters that satisfies each boundary value problem. However, the man- ifold of parameter sets to solve the problems for all measured pairs of P-I values of the curve is much smaller and can, with some carefully chosen external information, be reduced to a single solution. This is discussed next.
As discussed earlier, a typical P-I curve for a laser is a linear function of the current I, intersecting the I-axis at the threshold current Ith. When measured carefully, near the
threshold one finds a small bend, the “knee” of the curve. Two pieces of information can be determined from the linear function, the threshold and the slope, and a third from the curvature of the knee. This can fix three parameters. But since there are 6 parameters to be determined, the problem is obviously still under-determined. Thus the best way to determine all parameters is by first fitting the P-I curve of the uncoated laser, since then two parameters, the as-cleaved reflectivities of the laser facets, are known to be R1 = R2 ≈0.28.
If only one more parameter is determined through external measurement – e.g. αw that can
be calculated by measuring the threshold current as a function of laser length – the problem is fully determined and the remaining parameters, ¯g, ϕs(ν), ˆξsp will be stably found through
fitting.
on facet 1. Thus, with the other parameters given, the P-I curves (both facets) of the coated
laser can now be fitted with only R1 open. The value R1 fits to is the most accurate value
we can establish for the quality of the coating. Especially when R1 ≈ 0, it is important to
determine the residual reflectivity through this method due to the problems stated in Section 2.5.3.